Using Hull-White Interest Rate Trees

The Hull-White interest rate tree-building procedure was first outlined in the Fall 1994 issue of the Journal of Derivatives. It is becoming widely used by practitioners. This procedure is appropriate for models where there is some function x = f(r) of the short rate r that follows a mean- reverting arithmetic process. It can be used to implement the Ho-Lee model, the Hull-White model, and the Black- Karasinski model. Also, it is a tool that can be used for developing a wide range of new models. In this article we provide more details on the ways Hull- White trees can be used. We discuss the analytic results available when x = r, and make the point that it is important to distinguish between the per-period rate over one time step on the tree and the instantaneous short rate that is used in some of these analytic results. We provide an example of the implementation of the model using market data. We show how the tree can be designed so that it provides an exact fit to the initial volatility environment (but at the same time explain why we do not recommend this approach). We also discuss how to deal with such issues as variable time steps, cash flows that occur between nodes, barrier options, and path-dependence